Vectors & Scalars

Vector Addition A scalar is a quantity that has magnitude only Example: 4 miles A vector has both magnitude and direction Example: 4 miles north

Vectors are represented by arrows The direction of the arrow represents the direction of the vector and the length of the arrow represents the magnitude

Vectors may be added by the Tip-To-Tail Method Example: Four miles east plus three miles north The additive is called the Resultant, and can be calculated using the Pythagorean Theorem.

Try these: 5 m/s E and 10 m/s N 5 m/s E and 10 m/s W 5 m/s N and 10 m/s E

Parallel Transport of Vectors Vectors have a property known as parallel transport. The arrow representing a vector can be moved from one point to another, and as long as the length of the arrow and the direction of the arrow are unchanged, it is still the same vector.

Adding Three Vectors by the Tip-To-Tail Method Place the vectors tip-to-tail The sum of the vectors is the vector that goes from the tail of the first vector to the tip of the last vector A+B+C A The order of addition does not matter! A+B+C = B+C+A Vector Addition is COMMUTATIVE!

To subtract a vector: Add its negative. B B - A A -A Get the negative of a vector by reversing its direction. B B - A A Parallel transport the vectors to form a parallelogram -A Form the resultant vector: B + (-A) = B - A

Another way: Law of Sines sinA = sinB = sinC a b c There is also the Law of Cosines, but most students find the Law of Sines easier to work with!

Component Vectors Any vector can be split into component vectors. Cx Adding the component vectors gives the same result as adding the original vectors. C Cy The vector C has component vectors Cx and Cy. B A Ay By Cy = Ay + By Bx Ax Cx = Ax + Bx

Components vs. Component Vectors The x-component of a vector is (plus or minus) the length of the component vector Ax. The component is negative if the component vector is in the negative x-direction. (On the previous slide, we get the length of Cx by subtracting the length of Bx from the length of Ax. We can still write Cx = Ax + Bx, though, because Bx is negative.) Similarly, the y-component of a vector is (plus or minus) the length of the component vector Ay. The component is negative if the component vector is in the negative y-direction. NOTE: Usually the component vectors are written in bold and the components are not.

Calculating Vector Components Ay  Ax Ay sin  = opposite / hypotenuse = A Ax cos  = adjacent / hypotenuse = A Due to the way sine and cosine are defined for angles greater than 90º, if the angle is measured relative to the positive x-axis, the components will always have the correct sign when these equations are used.

Use the following technique: What if you don’t want to measure the angle relative to the positive x-axis? Use the following technique: What if I know the other angle? Ax Enclose the known angle with the component vectors. Ay A  Ay has length A cos  because Ay is adjacent to . Ay is positive because it is in the positive y-direction Ay has length A sin  because Ay is opposite to . Ay is positive because it is in the positive y-direction Ax has length A cos  because Ax is adjacent to . Ax is negative because it is in the negative x-direction Ax has length A sin  because Ax is opposite . Ax is negative because it is in the negative x-direction

We already know how to calculate Ax and Ay from A and . We may specify a (two-dimensional) vector in one of two ways: We give its components, Ax and Ay. Example: Ax = 4 miles east and Ay = 3 miles north We call this Rectangular Form 5 3 37º We give the magnitude and direction of the vector, A and  In this example A = 5 miles and  = 37º We call this Polar Form 4 The two forms are completely equivalent! We already know how to calculate Ax and Ay from A and . What about the reverse?

Calculating Magnitude and Direction of a Vector from its Components Ay  Ax Pythagorian Theorem: 2 A =  Ax2 + Ay2 -1 ( ) tan  = opposite / adjacent = Ay / Ax

CAUTION! There is a possible ambiguity with the inverse tangent! Because the tangent function repeats every 180º [i.e., tan( + 180º) = tan ], the inverse tangent function on your calculator will return results between - 90º and + 90º! SOLUTION: If the vector is in the second or third quadrant (when Ax is negative) ADD 180º TO THE RESULT GIVEN BY YOUR CALCULATOR